Problem 48 - Precalculus: Functions and Graphs

Question

Solve the equation or inequality. Express the solutions in terms of intervals whenever possible. $$-\frac{1}{2}<\frac{2 x+3}{5}<\frac{3}{2}$$

Step-by-Step Solution

Answer

\(x \in \left( -\frac{11}{4}, \frac{9}{4} \right)\)

Step 1: Understand the Compound Inequality

We are given a compound inequality: \(-\frac{1}{2}<\frac{2x+3}{5}<\frac{3}{2}\). This means we need to find the range of values of \(x\) that satisfy both inequalities simultaneously.

Step 2: Solve the First Inequality

Solve \(-\frac{1}{2}<\frac{2x+3}{5}\). Multiply both sides by 5 to eliminate the fraction:\(-\frac{1}{2} \times 5 < 2x + 3\).This simplifies to \(-\frac{5}{2} < 2x + 3\). Subtract 3 from both sides, resulting in \(-\frac{5}{2} - 3 < 2x\). This simplifies to \(-\frac{11}{2} < 2x\). Divide both sides by 2 to get \(-\frac{11}{4} < x\).

Step 3: Solve the Second Inequality

Solve \(\frac{2x+3}{5}<\frac{3}{2}\). Multiply both sides by 5:\(\frac{2x+3}{5} \times 5 < \frac{3}{2} \times 5\).This simplifies to \(2x + 3 < \frac{15}{2}\). Subtract 3 from both sides:\(2x < \frac{15}{2} - 3\) or \(2x < \frac{9}{2}\). Divide both sides by 2 to get \(x < \frac{9}{4}\).

Step 4: Combine Solutions

Combine the results from Step 2 and Step 3. We have two inequalities: \(-\frac{11}{4} < x < \frac{9}{4}\). This represents the interval:\(x \in \left( -\frac{11}{4}, \frac{9}{4} \right)\).

Key Concepts

Solving Inequalities

Solving inequalities is similar to solving equations, but instead of searching for a single value, we aim to find a set or range of values that satisfy the given conditions. When dealing with inequalities, different rules may apply compared to equations. For instance, when we multiply or divide an inequality by a negative number, we must remember to flip the inequality sign.
To solve a compound inequality like \(-\frac{1}{2}<\frac{2x+3}{5}<\frac{3}{2}\), we treat it as two separate inequalities:

• \(-\frac{1}{2}<\frac{2x+3}{5}\)
• \(\frac{2x+3}{5}<\frac{3}{2}\)

The objective is to solve each inequality independently and identify the common range for the variable \(x\). This involves various algebraic manipulations such as adding, subtracting, multiplying, or dividing terms to isolate \(x\). Once each inequality is solved, the solutions are combined based on their overlap, representing the full solution to the compound inequality.

Interval Notation

Interval notation is a mathematical method used to describe a range of values, usually solutions to an inequality. It provides a concise way to express the set of all possible solutions. In the context of the problem above, once we find that \(-\frac{11}{4} < x < \frac{9}{4}\), we can express it in interval notation as \(x \in \left( -\frac{11}{4}, \frac{9}{4} \right)\).
Interval notation uses parentheses, \(( )\), to denote that the endpoints are not included, and brackets, \([ ]\), to indicate that the endpoints are included. In our compound inequality, parentheses are used because neither \(-\frac{11}{4}\) nor \(\frac{9}{4}\) are part of the solution set.

• \((a, b)\) denotes all numbers between \(a\) and \(b\), but not including \(a\) and \(b\).
• \([a, b]\) represents all numbers between \(a\) and \(b\), including \(a\) and \(b\).

Thus, mastery of interval notation provides clarity and precision in expressing solutions to inequalities.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions. This skill is essential for solving equations and inequalities. It involves a range of techniques such as:

• Adding or subtracting the same amount from both sides of an equation or inequality.
• Multiplying or dividing both sides by a positive number, remembering the change of inequality direction if dealing with a negative number.
• Distributive property to eliminate brackets and grouping terms strategically.

In solving inequalities like \(-\frac{1}{2}<\frac{2x+3}{5}\), you might start by eliminating fractions through multiplication. Then, progressively isolate the variable, \(x\), using addition or subtraction, and further adjust by division to finalize the solution. These steps streamline the path to uncovering the solution set, enabling us to articulate it clearly, often using interval notation as the final step.